Problem 4 - University Physics with Modern Physics

Question

Sound having frequencies above the range of human hearing (about 20,000 Hz) is called \(ultrasound\). Waves above this frequency can be used to penetrate the body and to produce images by reflecting from surfaces. In a typical ultrasound scan, the waves travel through body tissue with a speed of 1500 m/s. For a good, detailed image, the wavelength should be no more than 1.0 mm. What frequency sound is required for a good scan?

Step-by-Step Solution

Answer

A frequency of 1.5 MHz is required for a good ultrasound scan.

Step 1: Understand the Relationship Between Speed, Frequency, and Wavelength

The speed of a wave, its frequency, and its wavelength are related by the equation \( v = f \lambda \), where \( v \) is the speed of the wave, \( f \) is the frequency, and \( \lambda \) is the wavelength.

Step 2: Identify the Known Quantities

We know the speed of the ultrasound wave in the body tissue is 1500 m/s and the wavelength for a good image is 1.0 mm, which is 0.001 meters.

Step 3: Rearrange the Formula to Solve for Frequency

We need to find the frequency, \( f \). Rearrange the wave equation to \( f = \frac{v}{\lambda} \).

Step 4: Substitute the Values into the Equation

Substitute the known values into the rearranged equation: \( f = \frac{1500}{0.001} \).

Step 5: Calculate the Frequency

Perform the calculation: \( f = \frac{1500}{0.001} = 1,500,000 \) Hz or 1.5 MHz.

Key Concepts

Wave Speed

Wave speed is an essential concept in understanding how waves move through different mediums. It indicates how fast a wave travels from one point to another. Wave speed (\( v \)...), frequency (\( f \)...), and wavelength (\( \lambda \)...) are interconnected. The speed of a wave is calculated by multiplying the frequency by the wavelength: \[ v = f \times \lambda \].

• For ultrasound waves traveling through the body, the speed is determined by the medium – in this case, body tissue, where it typically measures about 1500 m/s.
• This speed is essential for accurately timing reflections during diagnostic imaging.

Understanding wave speed helps in calculating other important parameters crucial for ultrasound imaging and numerous other applications in science and technology.

Wavelength

Wavelength is the distance between consecutive points of the same phase on a wave, such as crest to crest or trough to trough. It is a crucial factor in determining the level of detail available in an image.

• For ultrasound imaging, shorter wavelengths (such as 1.0 mm or 0.001 meters) are preferred as they produce higher resolution images.
• These shorter wavelengths can penetrate deep into the body, providing detailed insights crucial for medical diagnoses.

By understanding wavelength, we can control the resolution of the images produced, leading to more accurate and detailed ultrasound scans.

Wave Frequency

Wave frequency refers to the number of wave cycles that pass through a given point per second. It is measured in hertz (Hz). In the field of ultrasound imaging, frequency is vitally important.

• A higher frequency results in a shorter wavelength, which in turn, improves image resolution.
• For example, in a typical ultrasound scan, to achieve a wavelength of 1.0 mm, a frequency of 1.5 MHz is necessary.

Using the formula \( f = \frac{v}{\lambda} \), we can calculate the frequency needed to achieve the desired image quality. Frequency control allows medical professionals to tailor diagnostic procedures to each patient's needs.

Ultrasound Imaging

Ultrasound imaging is a non-invasive diagnostic tool used widely in the medical field. It uses high-frequency sound waves beyond the range of human hearing to create images of the inside of the body.

• Ultrasound waves are emitted by a probe and travel through body tissues, reflecting off structures and returning to the probe to form an image.
• Image clarity and detail depend on controlling wave speed, frequency, and wavelength.
• Typically, the wave speed in bodily tissues is around 1500 m/s, crucial for timing reflections correctly.

Understanding the principles behind ultrasound imaging helps in improving the diagnosis and treatment of various medical conditions, enhancing patient care.